Initial-boundary value problem for a class of linear relaxation systems in arbitrary space dimensions
type: Item
instance of: scholarly article
title: Initial-boundary value problem for a class of linear relaxation systems in arbitrary space dimensions
zbMATH Open document ID: 1011.35091
author: Wen-Qing Xu
published in: Journal of Differential Equations
publication date: 2002-11-28T00:00:00Z
review text: The author studies the behaviour of solutions of the following initial-boundary value problem in the half-space \(x_1\geq 0\), \(x'=(x_2,\dots,x_n)\in \mathbb{R}^{n-1}\) \[ \begin{cases} \frac{\partial u^{\varepsilon}}{\partial t}+ \sum_{k=1}^{n}\frac{\partial v_k^{\varepsilon}}{\partial x_k} =0,\\ \frac{\partial v_j^{\varepsilon}}{\partial t} +a_j\frac{\partial u^{\varepsilon}}{\partial x_j}= -\frac{1}{\varepsilon}(v^{\varepsilon}_j-\lambda_ju^{\varepsilon}), \quad j=1,\dots,n, \end{cases} \tag{1} \] \[ u^{\varepsilon}(x,0)=u_0(x),\quad v^{\varepsilon}_j(x,0)=v_{j0}(x), \quad j=1,\dots,n, \tag{2} \] \[ B_0u^{\varepsilon}(0,x',t)+B_1v_1^{\varepsilon}(0,x',t)=b(x',t) \tag{3} \] as \(\varepsilon \to 0\), where \(n\geq 2\), \(u^{\varepsilon}, v^{\varepsilon}, \lambda_j\in \mathbb{R}^n\), \(a_j>0\) and \(\varepsilon\) is a small positive parameter. The problem (1), (2), (3) will be called stiffly well posed if for any \(\alpha >0\), there exists a constant \(K=K(\alpha)>0\) independent on \(\varepsilon\), \(U_0\in L^2(\mathbb{R}^+\times \mathbb{R}^{n-1})\) and \(b\in L^2(\mathbb{R}^+\times \mathbb{R}^{n-1})\) such that \[ \begin{multlined} \int_0^{\infty}\int_{\mathbb{R}^n_+}|U^{\varepsilon}(x,t)|^2e^{-2\alpha t} dx dt+ \int_0^{\infty}\int_{\mathbb{R}^{n-1}_{+}}|U^{\varepsilon}(0,x',t)|^2e^{-2\alpha t} dx' dt+\\ \leq K\Big(\int_0^{\infty}\int_{\mathbb{R}^{n-1}}|b(x',t)|^2e^{-2\alpha t} dx' dt+ \int_{\mathbb{R}^n_+}|U_0(x)|^2 dx\Big),\end{multlined} \] where \(U^{\varepsilon}=\text{col}(u^{\varepsilon},v^{\varepsilon}_1, \dots,v^{\varepsilon}_n)\). Assuming that \( \sum_{j=1}^n\lambda_j^2/a_j\leq 1 \) and \(U_0(x)\equiv 0\) the author proves that the problem (1),(2),(3) is stiffly well posed if and only if the boundary condition (3) satisfies the Uniform Kreiss Condition (UKC): \(B_0B_1>0\). If (UKC) holds and \(b\in L^2(\mathbb{R}^+\times \mathbb{R}^{n-1})\), then there exists a unique solution \(U= \text{col}(u,v_1,\dots,v_n)\) of the problem \[ \frac{\partial u}{\part…
reviewed by: Andrei Perjan
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1836803
zbMATH Keywords: hyperbolic systems with relaxation
zbMATH Keywords: stiff well posedness
zbMATH Keywords: asymptotic convergence
zbMATH Keywords: subcharacteristic condition
zbMATH Keywords: stiff Kreiss condition
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4130
OpenAlex ID: W2041222373
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DOI: 10.1006/JDEQ.2001.4130
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description: scientific article; zbMATH DE number 1836803